Question: Given $ \overrightarrow{PQ}\perp\overrightarrow{PS}$, $ m \angle RPS = 8x - 48$, and $ m \angle QPR = 7x - 42$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since we are given that $\overrightarrow{PQ}\perp\overrightarrow{PS}$ , we know ${m\angle QPS = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 42} + {8x - 48} = {90}$ Combine like terms: $ 15x - 90 = 90$ Add $90$ to both sides: $ 15x = 180$ Divide both sides by $15$ to find $x$ $ x = 12$ Substitute $12$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 8({12}) - 48$ Simplify: $ {m\angle RPS = 96 - 48}$ So ${m\angle RPS = 48}$.